Weyl-Gauging and Conformal Invariance

TL;DR

This paper explores the relationship between scale, Weyl, and conformal invariance in curved space, providing criteria for conformal invariance and linking Weyl anomalies to central extensions.

Contribution

It establishes a precise connection between Weyl-gauging and curvature coupling, offering a systematic way to identify conformally invariant actions and improved energy-momentum tensors.

Findings

Classifies actions where Weyl-gauging equals curvature coupling.

Provides an algebraic criterion for conformal invariance.

Connects Weyl anomalies with central extensions in 2D.

Abstract

Scale-invariant actions in arbitrary dimensions are investigated in curved space to clarify the relation between scale-, Weyl- and conformal invariance on the classical level. The global Weyl-group is gauged. Then the class of actions is determined for which Weyl-gauging may be replaced by a suitable coupling to the curvature (Ricci gauging). It is shown that this class is exactly the class of actions which are conformally invariant in flat space. The procedure yields a simple algebraic criterion for conformal invariance and produces the improved energy-momentum tensor in conformally invariant theories in a systematic way. It also provides a simple and fundamental connection between Weyl-anomalies and central extensions in two dimensions. In particular, the subset of scale-invariant Lagrangians for fields of arbitrary spin, in any dimension, which are conformally invariant is given. An example of a quadratic action for which scale-invariance does not imply conformal invariance is constructed.